Question

The triangle below is equilateral. Find the length of side a in simplest radical form

with a rational denominator.

Answer 1

looking at the triangle, we can see that segment "11" is coming from a vertex down to a perpendicular "base", anyhow, in short that means that the segment that's 11 units long is the altitude or height of the triangle and it's cutting the base in two equal halves.

`\textit{height of an equilateral triangle}\\\\ h=\cfrac{s√(3)}{2}~~ \begin{cases} s=\stackrel{length~of}{a~side}\\[-0.5em] \hrulefill\\ h=11 \end{cases}\implies 11=\cfrac{s√(3)}{2}\implies 22=s√(3) \\\\\\ \cfrac{22}{√(3)}=s\implies \cfrac{22}{√(3)}\cdot \cfrac{√(3)}{√(3)}=s\implies \cfrac{22√(3)}{3}=s~\hfill \stackrel{\textit{half of`